What's the truth about phase velocity vs. group velocity?

[peter.ell]

I know the phase vs group question has been asked before, but I really don't quite understand it. I've read in various places that the phase velocity of a light wave has been made to go faster than C, and in others that the group velocity has been made to go faster than C, and even made negative so that it appears to exit a medium before it even enters. This is called anomalous dispersion, but what in the world does this mean?

What, conceptually, is the difference between phase velocity and group velocity, which is the one that can really go faster than C, and how can it do so?

Refer to this for the source of my questions: http://en.wikipedia.org/wiki/Dispersion_(optics)#Group_and_phase_velocity

Thank you so much!

 

[jtbell]

Here's an applet that shows an example of group velocity greater than phase velocity.

http://gregegan.customer.netspace.net.au/APPLETS/20/20.html

This is actually rather unusual, and occurs under the conditions described on that page. Usually the group velocity is less than the phase velocity.

 

[chickenz]

Perhaps someone can tell the original poster the conditions for which the signal velocity (which must be less than c) is approximately equal to the phase velocity or to the group velocity or to neither.

I believe that, most of the time, if there is a fairly well-defined envelope for a wave, the group velocity is a good description of how fast information and energy propagates. And that the phase velocity should almost never be used as a signal velocity. Someone correct me if I am wrong.

 

[lightarrow]

What, conceptually, is the difference between phase velocity and group velocity

Simplistically speaking, phase velocity is the velocity at which a peak (or a trough) moves; group velocity is the velocity at which an entire group of waves moves.

which is the one that can really go faster than C

Both. Neither one nor the other can represent the "signal velocity", excepting for special situations.

and how can it do so?

I will use a metaphor: if you put many lamps in a row and you make them switch on in a specific temporal order, you can make a flash of light move along the row at the speed you want, even > c. But this doesn't correspond to a physical signal moving along the lamps, because the lamps were programmed to switch on at the specified timing.
You can do the same with phase velocity and with group velocity (as in the nice applet posted by jtbell).

 

[PJC01]

I can understand your confusion about the concept of phase velocity and group velocity. Let me try to explain it in simpler terms.

Phase velocity refers to the speed at which the phase of a wave travels. In other words, it is the speed at which the peaks and troughs of a wave move through space. On the other hand, group velocity refers to the speed at which the energy of a wave travels. It is the speed at which the envelope of a wave (which represents the energy of the wave) moves through space.

Now, the reason why the phase velocity and group velocity can be different is due to a phenomenon called dispersion. Dispersion occurs when different wavelengths of light travel at different speeds through a medium. This can happen because the refractive index of a medium (which determines the speed of light in that medium) can vary with wavelength.

In normal dispersion, the longer wavelengths of light travel slower than the shorter wavelengths, resulting in a positive group velocity. However, in anomalous dispersion, the opposite happens – the longer wavelengths travel faster than the shorter wavelengths, resulting in a negative group velocity. This means that the energy of the wave appears to travel backwards through space.

But it is important to note that neither the phase velocity nor the group velocity can actually exceed the speed of light in a vacuum (c). When we say that the group velocity can be faster than c, it means that the envelope of the wave (which represents the energy) appears to travel faster than c, but the actual speed of the individual wavelengths (represented by the phase velocity) is still less than c.

In summary, the difference between phase velocity and group velocity lies in the concept of dispersion and how different wavelengths of light travel through a medium. While both velocities can appear to exceed the speed of light in certain cases, they cannot actually do so. I hope this helps clarify your understanding of this topic.